In this document we reperform some of the analysis provided in Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure by Siddhartha R. Dalal, Edward B. Fowlkes, Bruce Hoadley published in Journal of the American Statistical Association, Vol. 84, No. 408 (Dec., 1989), pp. 945-957 and available at http://www.jstor.org/stable/2290069.
On the fourth page of this article, they indicate that the maximum likelihood estimates of the logistic regression using only temperature are: $\hat{\alpha}=5.085$ and $\hat{\beta}=-0.1156$ and their asymptotic standard errors are $s_{\hat{\alpha}}=3.052$ and $s_{\hat{\beta}}=0.047$. The Goodness of fit indicated for this model was $G^2=18.086$ with 21 degrees of freedom. Our goal is to reproduce the computation behind these values and the Figure 4 of this article, possibly in a nicer looking way.
We will be using the python3 language using the pandas, statsmodels, numpy, matplotlib and seaborn libraries.
def print_imported_modules():
import sys
for name, val in sorted(sys.modules.items()):
if(hasattr(val, '__version__')):
print(val.__name__, val.__version__)
# else:
# print(val.__name__, "(unknown version)")
def print_sys_info():
import sys
import platform
print(sys.version)
print(platform.uname())
print(platform.python_version())
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
import seaborn as sns
print_sys_info()
print_imported_modules()
3.7.16 (default, Jan 17 2023, 16:06:28) [MSC v.1916 64 bit (AMD64)] uname_result(system='Windows', node='CLT-PHAC2105-P', release='10', version='10.0.19041', machine='AMD64', processor='Intel64 Family 6 Model 140 Stepping 1, GenuineIntel') 3.7.16 IPython 7.31.1 IPython.core.release 7.31.1 PIL 9.3.0 PIL.Image 9.3.0 PIL._deprecate 9.3.0 PIL._version 9.3.0 _csv 1.0 _ctypes 1.1.0 decimal 1.70 _pydev_bundle.fsnotify 0.1.5 _pydevd_frame_eval.vendored.bytecode 0.13.0.dev argparse 1.1 backcall 0.2.0 bottleneck 1.3.5 cffi 1.15.1 colorama 0.4.6 csv 1.0 ctypes 1.1.0 cycler 0.10.0 dateutil 2.8.2 debugpy 1.5.1 decimal 1.70 decorator 5.1.1 defusedxml 0.7.1 entrypoints 0.4 http.server 0.6 ipykernel 6.15.2 ipykernel._version 6.15.2 ipython_genutils 0.2.0 ipython_genutils._version 0.2.0 ipywidgets 7.6.5 ipywidgets._version 7.6.5 jedi 0.18.1 json 2.0.9 jupyter_client 7.4.9 jupyter_client._version 7.4.9 jupyter_core 4.11.2 jupyter_core.version 4.11.2 kiwisolver 1.4.4 kiwisolver._cext 1.4.4 logging 0.5.1.2 matplotlib 3.5.3 mkl 2.4.0 numexpr 2.8.4 numpy 1.21.5 numpy.core 1.21.5 numpy.core._multiarray_umath 3.1 numpy.lib 1.21.5 numpy.linalg._umath_linalg 0.1.5 packaging 22.0 packaging.__about__ 22.0 pandas 1.3.5 parso 0.8.3 patsy 0.5.6 patsy.version 0.5.6 pickleshare 0.7.5 pkg_resources._vendor.appdirs 1.4.3 pkg_resources._vendor.more_itertools 8.12.0 pkg_resources._vendor.packaging 21.3 pkg_resources._vendor.packaging.__about__ 21.3 pkg_resources._vendor.pyparsing 3.0.9 pkg_resources._vendor.appdirs 1.4.3 pkg_resources._vendor.more_itertools 8.12.0 pkg_resources._vendor.packaging 21.3 pkg_resources._vendor.pyparsing 3.0.9 platform 1.0.8 prompt_toolkit 3.0.36 psutil 5.9.0 pydevd 2.6.0 pygments 2.11.2 pyparsing 3.0.9 pytz 2022.7 re 2.2.1 scipy 1.7.3 scipy._lib._uarray 0.5.1+49.g4c3f1d7.scipy scipy._lib.decorator 4.0.5 scipy.integrate._dop b'$Revision: $' scipy.integrate._ode $Id$ scipy.integrate._odepack 1.9 scipy.integrate._quadpack 1.13 scipy.integrate.lsoda b'$Revision: $' scipy.integrate.vode b'$Revision: $' scipy.interpolate._fitpack 1.7 scipy.interpolate.dfitpack b'$Revision: $' scipy.linalg._fblas b'$Revision: $' scipy.linalg._flapack b'$Revision: $' scipy.linalg._flinalg b'$Revision: $' scipy.linalg._interpolative b'$Revision: $' scipy.ndimage 2.0 scipy.optimize.__nnls b'$Revision: $' scipy.optimize._cobyla b'$Revision: $' scipy.optimize._lbfgsb b'$Revision: $' scipy.optimize._minpack 1.10 scipy.optimize._slsqp b'$Revision: $' scipy.optimize.minpack2 b'$Revision: $' scipy.signal.spline 0.2 scipy.sparse.linalg.eigen.arpack._arpack b'$Revision: $' scipy.sparse.linalg.isolve._iterative b'$Revision: $' scipy.special.specfun b'$Revision: $' scipy.stats.mvn b'$Revision: $' scipy.stats.statlib b'$Revision: $' seaborn 0.12.2 seaborn.external.appdirs 1.4.4 seaborn.external.husl 2.1.0 six 1.16.0 socketserver 0.4 statsmodels 0.13.5 statsmodels.__init__ 0.13.5 statsmodels._version 0.13.5 statsmodels.api 0.13.5 statsmodels.tools.web 0.13.5 traitlets 5.7.1 traitlets._version 5.7.1 urllib.request 3.7 wcwidth 0.2.5 xmlrpc.client 3.7 zlib 1.0 zmq 23.2.0 zmq.sugar 23.2.0 zmq.sugar.version 23.2.0
Main libraries :
Python version : 3.7.16
platform : Jupyter
OS : Windows x64 (build 19045)
Let's start by reading data.
data = pd.read_csv("https://app-learninglab.inria.fr/moocrr/gitlab/moocrr-session3/moocrr-reproducibility-study/raw/master/data/shuttle.csv") # Modify "blob to raw"
data
| Date | Count | Temperature | Pressure | Malfunction | |
|---|---|---|---|---|---|
| 0 | 4/12/81 | 6 | 66 | 50 | 0 |
| 1 | 11/12/81 | 6 | 70 | 50 | 1 |
| 2 | 3/22/82 | 6 | 69 | 50 | 0 |
| 3 | 11/11/82 | 6 | 68 | 50 | 0 |
| 4 | 4/04/83 | 6 | 67 | 50 | 0 |
| 5 | 6/18/82 | 6 | 72 | 50 | 0 |
| 6 | 8/30/83 | 6 | 73 | 100 | 0 |
| 7 | 11/28/83 | 6 | 70 | 100 | 0 |
| 8 | 2/03/84 | 6 | 57 | 200 | 1 |
| 9 | 4/06/84 | 6 | 63 | 200 | 1 |
| 10 | 8/30/84 | 6 | 70 | 200 | 1 |
| 11 | 10/05/84 | 6 | 78 | 200 | 0 |
| 12 | 11/08/84 | 6 | 67 | 200 | 0 |
| 13 | 1/24/85 | 6 | 53 | 200 | 2 |
| 14 | 4/12/85 | 6 | 67 | 200 | 0 |
| 15 | 4/29/85 | 6 | 75 | 200 | 0 |
| 16 | 6/17/85 | 6 | 70 | 200 | 0 |
| 17 | 7/2903/85 | 6 | 81 | 200 | 0 |
| 18 | 8/27/85 | 6 | 76 | 200 | 0 |
| 19 | 10/03/85 | 6 | 79 | 200 | 0 |
| 20 | 10/30/85 | 6 | 75 | 200 | 2 |
| 21 | 11/26/85 | 6 | 76 | 200 | 0 |
| 22 | 1/12/86 | 6 | 58 | 200 | 1 |
We know from our previous experience on this data set that filtering data is a really bad idea. We will therefore process it as such.
%matplotlib inline
pd.set_option('mode.chained_assignment',None) # this removes a useless warning from pandas
import matplotlib.pyplot as plt
data["Frequency"]=data.Malfunction/data.Count
data.plot(x="Temperature",y="Frequency",kind="scatter",ylim=[0,1])
plt.grid(True)
Let's assume O-rings independently fail with the same probability which solely depends on temperature. A logistic regression should allow us to estimate the influence of temperature.
import statsmodels.api as sm
data["Success"]=data.Count-data.Malfunction
data["Intercept"]=1
logmodel=sm.GLM(data['Frequency'],
data[['Intercept','Temperature']],
family=sm.families.Binomial(sm.families.links.logit())).fit() # Added "()" after "logit"
logmodel.summary()
| Dep. Variable: | Frequency | No. Observations: | 23 |
|---|---|---|---|
| Model: | GLM | Df Residuals: | 21 |
| Model Family: | Binomial | Df Model: | 1 |
| Link Function: | logit | Scale: | 1.0000 |
| Method: | IRLS | Log-Likelihood: | -3.9210 |
| Date: | Thu, 22 Aug 2024 | Deviance: | 3.0144 |
| Time: | 09:30:41 | Pearson chi2: | 5.00 |
| No. Iterations: | 6 | Pseudo R-squ. (CS): | 0.04355 |
| Covariance Type: | nonrobust |
| coef | std err | z | P>|z| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| Intercept | 5.0850 | 7.477 | 0.680 | 0.496 | -9.570 | 19.740 |
| Temperature | -0.1156 | 0.115 | -1.004 | 0.316 | -0.341 | 0.110 |
The maximum likelyhood estimator of the intercept and of Temperature are thus $\hat{\alpha}=5.0850$ and $\hat{\beta}=-0.1156$. This corresponds to the values from the article of Dalal et al. The standard errors are $s_{\hat{\alpha}} = 7.477$ and $s_{\hat{\beta}} = 0.115$, which is different from the $3.052$ and $0.04702$ reported by Dallal et al. The deviance is $3.01444$ with 21 degrees of freedom. I cannot find any value similar to the Goodness of fit ($G^2=18.086$) reported by Dalal et al. There seems to be something wrong. Oh I know, I haven't indicated that my observations are actually the result of 6 observations for each rocket launch. Let's indicate these weights (since the weights are always the same throughout all experiments, it does not change the estimates of the fit but it does influence the variance estimates).
logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']],
family=sm.families.Binomial(sm.families.links.logit()), # Added "()" after "logit"
var_weights=data['Count']).fit()
logmodel.summary()
| Dep. Variable: | Frequency | No. Observations: | 23 |
|---|---|---|---|
| Model: | GLM | Df Residuals: | 21 |
| Model Family: | Binomial | Df Model: | 1 |
| Link Function: | logit | Scale: | 1.0000 |
| Method: | IRLS | Log-Likelihood: | -23.526 |
| Date: | Thu, 22 Aug 2024 | Deviance: | 18.086 |
| Time: | 09:30:41 | Pearson chi2: | 30.0 |
| No. Iterations: | 6 | Pseudo R-squ. (CS): | 0.2344 |
| Covariance Type: | nonrobust |
| coef | std err | z | P>|z| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| Intercept | 5.0850 | 3.052 | 1.666 | 0.096 | -0.898 | 11.068 |
| Temperature | -0.1156 | 0.047 | -2.458 | 0.014 | -0.208 | -0.023 |
Good, now I have recovered the asymptotic standard errors $s_{\hat{\alpha}}=3.052$ and $s_{\hat{\beta}}=0.047$. The Goodness of fit (Deviance) indicated for this model is $G^2=18.086$ with 21 degrees of freedom (Df Residuals).
I have therefore managed to fully replicate the results of the Dalal et al. article.
The temperature when launching the shuttle was 31°F. Let's try to estimate the failure probability for such temperature using our model.:
%matplotlib inline
data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121), 'Intercept': 1})
data_pred['Frequency'] = logmodel.predict(data_pred)
data_pred.plot(x="Temperature",y="Frequency",kind="line",ylim=[0,1])
plt.scatter(x=data["Temperature"],y=data["Frequency"])
plt.grid(True)
This figure is very similar to the Figure 4 of Dalal et al. I have managed to replicate the Figure 4 of the Dalal et al. article.
Following the documentation of Seaborn, I use regplot.
sns.set(color_codes=True)
plt.xlim(30,90)
plt.ylim(0,1)
sns.regplot(x='Temperature', y='Frequency', data=data, logistic=True)
plt.show()
I think I have managed to correctly compute and plot the uncertainty of my prediction. Although the shaded area seems very similar to the one obtained by with R, I can spot a few differences (e.g., the blue point for temperature 63 is outside)... Could this be a numerical error ? Or a difference in the statistical method ? It is not clear which one is "right".